Transient Pressure Analysis of Pressure Sensitive Naturally Fractured Reservoirs

This paper presents the results of a study with the main aim of investigating the effect of considering the pressure sensitivity of the natural fractures of a fissured formation on the transient pressure analysis of well tests. Two motivations for this study were as follows. First, many (or most in some cases) of the producing reservoirs (geothermal and petroleum) are naturally fractured, showing a stress sensitivity behavior of the fractures. Second, some reservoir simulators consider the option of varying the fractured media properties with the exploitation time. This work discusses a mathematical model for the radial flow of a fluid through a pressure sensitive naturally fractured formation, with pseudosteady state matrix-fracture flow as described by Warren and Root. This model includes the effects of wellbore storage, a finite skin (damage) around the wellbore and high-velocity flow. A finite difference solution for this model is also briefly described. Simulated examples of drawdown tests are analized. INTRODUCTION It has been recognized that porous media are not always, rigid and non deformable. This problem is ussually handled by means of properly chosen “average” properties. This method only reduces the errors involved and generally and does not totally eliminate them. This is especially true for naturally fractured reservoirs. A review of the well testing literature indicates that this variable rock properties problem for naturally fractured systems has not been addressed. In other words, the published treatments solve the diffusivity type coupled equations for the fractured and matrix systems, for homogeneous flow, assuming that the diffusivity is a constant independent of pressure. When both pressure and property changes of both systems are small, the constant property assumption is justified. If instead, rock and fluid properties changes are important over the pressure range of interest, these changes cannot be neglected and a variable property solution should be obtained. The motivations for this work were basically two. The first was based on the fact that many hydrocarbon reservoirs and most geothermal reservoirs are naturally fractured, and it has been shown through petrophysical studies (Jones, 1975) and well testing (Adams, 1983), that the permeability of the fractures is a function of the effective stress (in a simplified manner defined as the difference between the external or overburden stress and the pore pressure). Whenever a formation presents this problem, it is necessary to quantitatively assess the impact of its stress sensitivity characteristics on the producing conditions of the wells. The second motivation was based on the option of some commercial reservoir simulators, that consider the variation of the naturally fractured systems during its exploitation (Peng et al., 1983; GeoQuest, 1998). The main aim of this paper is to present the results of a study that considers the pressure dependency of the fracture properties. Implicit in this work is the assumption of a linear-elastic medium with no hysterisis. Results are presented for drawdow tests, in terms of a psudopressure pfD p . The investigation also includes the effects of wellbore damage, wellbore storage and high-velocity flow. MATHEMATICAL FORMULATION To formulate the mathematical model, the assumptions ussually made in well testing theory are applied. We assume horizontal flow with no gravity effects, a fully penetrating well, isothermal singlephase fluid obeying Forchheimer equation, an isotropic and homogeneous formation, and purely elastic fracture properties. The assumption of horizontal flow is not quite valid because of changes in porosity and height with pressure. However, it is easily shown (Samaniego,